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authorGravatar Hugo Herbelin <Hugo.Herbelin@inria.fr>2016-08-18 13:24:50 +0200
committerGravatar Hugo Herbelin <Hugo.Herbelin@inria.fr>2016-09-26 19:47:04 +0200
commit980b434552d73cb990860f8d659b64686f6dbc87 (patch)
tree11f5a096cdee39bed6bf8b7e8cf6bee0e5ffd02e /test-suite/modules/Nat.v
parent5d36de8e2304661e76ac433d5dd90c13b3b86c72 (diff)
Inference of return clause: giving uniformly priority to "small inversion".
As noted by Jason Gross on coq-club (Aug 18, 2016), the "small inversion" heuristic is not used consistently depending on whether the variables in the type constraint are Rel or Var. This commit simply gives uniformly preference to the inversion of the predicate along the indices of the type over other heuristics. The next three commits will improve further a uniform use of the different heuristics. ---------------------------------------------------------------------- Here are some extra comments on how to go further with the inference of the return predicate: The "small inversion" heuristic build_inversion_problem (1) is characterized by two features: - small inversion properly speaking (a), i.e. that is for a match on t:I params p1(u11..u1p1) ... pn(un1..unpn) with pi exposing the constructor structure of the indices of the type of t, a return clause of the form "fun x1..xn (y:I params x1..xn) => match x1..xn y with | p1(z11..z1p1) ... pn(zn1..znpn) => ?T@{z11..znpn} | _ => IDProp end" is used, - the dependent subterms in the external type constraint U are replaced by existential variables (b) which can be filled either by projecting (i.e. installing a dependency) or imitating (i.e. no dependency); this is obtained by solving the constraint ?T@{u11..unpn} == U by setting ?T@{z11..znpn} := U'(...?wij@{zij:=uij}...) where U has been written under the form U'(...uij...) highlighting all occurrences of each of the uij occurring in U; otherwise said the problem is reduced to the question of instantiating each wij, deciding whether wij@{zij} := zij (projection) or wij@{zij} := uij (imitation) [There may be different way to expose the uij in U, e.g. in the presence of overlapping, or of evars in U; this is left undetermined]. The two other heuristics used are: - prepare_predicate_from_arsign_tycon (2): takes the external type constraint U and decides that each subterm of the form xi or y for a match on "y:I params x1 ... xn" is dependent; otherwise said, it corresponds to the degenerated form of (1) where - no constructor structure is exposed (i.e. each pi is trivial) - only uij that are Rel are replaced by an evar ?wij and this evar is directly instantiated by projection (hence creating a dependency), - simple use of of an evar in case no type constraint is given (3): this evar is not dependent on the indices nor on the term to match. Heuristic (1) is not strictly more powerful than other heuristics because of (at least) two weaknesses. - The first weakness is due to feature (b), i.e. to letting unification decide whether these evars have to create a dependency (projection) or not (imitation). In particular, the heuristic (2) gives priority to systematic abstraction over the dependencies (i.e. giving priority to projection over imitation) and it can then be better as the following example (from RelationClasses.v) shows: Fixpoint arrows (l : Tlist) (r : Type) : Type := match l with | Tnil => r | A :: l' => A -> arrows l' r end. Fixpoint predicate_all (l : Tlist) : arrows l Prop -> Prop := match l with | Tnil => fun f => f | A :: tl => fun f => forall x : A, predicate_all tl (f x) end. Using (1) fails. It proposes the predicate "fun l' => arrows ?l[l':=l'] Prop" so that typing the first branch leads to unify "arrows ?l[l:=Tnil] Prop == Prop", a problem about which evarconv unification is not able (yet!) to see what are the two possible solutions. Using (2) works. It instead directly suggests that the predicate is "fun l => arrows l Prop" is used, so that unification is not needed. Even if in practice the (2) is good (and hence could be added to (1)), it is not universally better. Consider e.g. y:bool,H1:P y,H2:P y,f:forall y, P y -> Q y |- match y as z return Q y with | true => f y H1 | false => f y H2 end : Q y There is no way to type it with clause "as z return Q z" even if trying to generalize H1 and H2 so that they get type P z. - A second weakness is due to the interaction between small inversion and constructors having a type whose indices havex a less refined constructor structure than in the term to match, as in: Inductive I : nat -> Set := | C1 : forall n : nat, listn n -> I n | C2 : forall n : nat, I n -> I n. Check (fun x : I 0 => match x with | C1 n l => 0 | C2 n c => 0 end). where the inverted predicate is "in I n return match n with 0 => ?T | _ => IDProp end" but neither C1 nor C2 have fine enough types so that n becomes constructed. There is a generic solution to that kind of situation which is to compile the above into Check (fun x : I 0 => match x with | C1 n l => match n with 0 => 0 | _ -> id end | C2 n c => match n with 0 => 0 | _ -> id end end). but this is not implemented yet. In the absence of this refinement, heuristic (3) can here work better. So, the current status of the claim is that for (1) to be strictly more powerful than other current heuristics, work has to be done - (A) at the unification level (by either being able to reduce problems of the form "match ?x[constructor] with ... end = a-rigid-term", or, at worst, by being able to use the heuristic favoring projecting for such a problem), so that it is better than (2), - (B) at the match compilation level, by enforcing that, in each branch, the corresponding constructor is refined so has to match (or discriminate) the constraints given by the type of the term to match, and hence being better than (3). Moreover, (2) and (3) are disjoint. Here is an example which (3) can solve but not (2) (and (1) cannot because of (B)). [To be fixed in next commit.] Inductive I : bool -> bool -> Type := C : I true true | D x : I x x. Check fun z P Q (y:I true z) (H1 H2:P y) (f:forall y, P y -> Q y z) => match y with | C => f y H1 | D _ => f y H2 end : Q y z. Indeed, (2) infers "as y' in I b z return Q y z" which does not work. Here is an example which (2) can solve but not (3) (and (1) cannot because of (B) again). [To be fixed in 2nd next commit]. Check fun z P Q (y:I true z) (H1 H2:P y) (f:forall y z, P y -> Q y z) => match y with | C => f y true H1 | D b => f y b H2 end : Q y z.
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